Positivity of the height jump divisor

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by  Jose Ignacio Burgos Gil

POSITIVITY OF THE HEIGHT JUMP DIVISOR
JOSE´ IGNACIOBURGOSGIL,DAVIDHOLMES,ANDROBINDEJONG
7
1
0 Abstract. Westudythedegenerationofsemipositivesmoothhermitianline
2 bundles on open complex manifolds, assuming that the metric extends well
n away from a codimension two analytic subset of the boundary. Using ter-
a minology introduced by R. Hain, we show that under these assumptions the
J so-calledheightjumpdivisorsarealwayseffective. Thisresultisofparticular
interest in the context of biextension line bundles on Griffiths intermediate
2
jacobian fibrations of polarized variations of Hodge structure of weight −1,

pulledbackalongnormalfunctionsections. Inthecaseofthenormalfunction
]
V onMg associatedtotheCeresacycle,ourresultprovesaconjectureofHain.
C As an application of our result we obtain that the Moriwaki divisor on Mg
has non-negative degree on all complete curves inMg not entirely contained
h. inthelocusofirreduciblesingularcurves.
t
a
m
[

1. Introduction
1
v Let X be a complex manifold, E a reduced divisor on X, and set U = X \|E|.
0
LetLbeaholomorphiclinebundleonU equippedwithasmoothhermitianmetric
7
k· k. The purpose of this note is to show a certain concavity property for the
3
singularities of the metric k·k across the boundary divisor E, provided (a) the
0
0 metric is semipositive over U, and (b) the metric extends continuously away from
. an analytic subset S ⊂ |E| of codimension at least 2 in X. Here we say that the
1
0 metric k·k is semipositive if the first Chern form c1(L) of L = (L,k·k) is semi-
7 positiveasa(1,1)-formonU. AtypicalexampleofananalyticsetS asincondition
1 (b) would be the singular locus of |E|.
:
v ToformulatetheconcavitypropertyitisusefultointroducethenotionofaLear
i extension, following R. Hain [12, Sections 6 and 14], inspired by the results of D.
X
Lear in [18]. We say that the smooth hermitian line bundle L is Lear-extendable
r
a overX ifthereexistsananalyticsubsetS ⊂|E|ofcodimensionatleast2inX and
⊗N
an integer N ∈Z such that the smooth hermitian line bundle L extends as a
>0
continuously metrized holomorphic line bundle over X \S. If we assume that the
metrick·kissemipositiveinU,thentheassumptionthatS isananalyticsubsetof
codimensionatleast2 inX implies that the holomorphicline bundle L⊗N extends
(uniquely)asaholomorphicline bundleonthe wholeofX,by[23,Theorem1]and
[25, Proposition 2].
Thus, if L is Lear-extendable over X and the metric k ·k is semipositive on
U, then the line bundle L has a canonical extension as a Q-line bundle over X,
The first author is partiallysupported by the MINECO research project MTM2013-42135-P,
theICMATSeveroOchoaprojectSEV-2015-0554andtheDFGprojectSFB1085“HigherInvari-
ants”. The second and thirdauthors would liketo thank the Instituto de Ciencias Matem´eticas
(ICMAT)forgenerouslyhostinguswhilemuchofthisresearchwascarriedout.
1
2 JOSE´ IGNACIOBURGOSGIL,DAVIDHOLMES,ANDROBINDEJONG
determined by the chosenmetric k·k on L. We denote this canonicalextension by
[L,X] and call it the Lear extension of L.
LetC beaconnectedRiemannsurfaceandletϕ: C →X beaholomorphicmap
such that the generic point of C has image in U. Let V =ϕ−1U, which is an open
dense subset of C. If we assume that L is Lear-extendable over X and in addition
that ϕ|∗L is Lear-extendable over C, it is natural to consider the difference of
V
Q-line bundles
ϕ∗[L,X]−[ϕ|∗ L,C]
V
onC. Sinceϕ∗[L,X]and[ϕ|∗L,C]agreeonV, thedifference ϕ∗[L,X]−[ϕ|∗L,C]
V V
determinescanonicallyaQ-divisorsupportedonthe boundarysetC\V, whichwe
write as J = J . Following terminology introduced by Hain [12, Section 14] we
ϕ,L
call J the height jump divisor of L with respect to the morphism ϕ. We say that
the height jumps if J is non-zero. Note that the height jump divisor J measures
the failure of the Lear extension to be compatible with pullback along ϕ.
Our main aim in this note is to prove the following result.
Theorem 1.1. Assume that the metric of the smooth hermitian holomorphic line
bundle L on U is semipositive. Moreover, assume that L has a Lear extension
[L,X] over X, and that ϕ|∗L has a Lear extension [ϕ|∗L,C] over C. Then the
V V
height jump divisor J on C is effective.
ϕ,L
The main motivation for this result comes from workof Hain [12], G. Pearlstein
[22],P.BrosnanandPearlstein[4]andPearlsteinandC.Peters[21]concerningthe
curvature properties and asymptotic behavior of admissible biextension variations
of mixed Hodge structure, in particular a conjecture of Hain formulated in [12,
Section 14].
We briefly sketch this circle of ideas, referring to [12] for more details. Let V
denote a polarized variation of Hodge structure of weight −1 over the complex
manifold U = X \ |E|. By classical work of P. Griffiths one has associated to
V a canonical torus bundle J(V) → U, called the intermediate jacobian fibration
of V over U. As is explained in [10], [12] the polarization on V gives rise to a
canonical holomorphic line bundle B on J(V) whose total space classifies self-dual
biextension variations of mixed Hodge structures of V. A self-dual biextension
variation of mixed Hodge structures of V is a variation of mixed Hodge structures
B over U with weight graded quotients
Gr ∼=Z(0), Gr ∼=V, Gr ∼=Z(1)
0 −1 −2
such that the resulting extensions W B/Z(1) ∈ Ext1 (Z(0),V) and W B ∈
0 MHS(U) −1
Ext1 (V,Z(1)) are identified via the canonical map
MHS(U)
Ext1 (Z(0),V)=J(V)(U)−→Jˇ(V)(U)=Ext1 (V,Z(1))
MHS(U) MHS(U)
determined by the polarization of V. The biextension line bundle B is equipped
with a canonical smooth hermitian metric k·k, and we write B =(B,k·k).
Letν: U →J(V)beaholomorphicsectionofthetorusbundleJ(V)→U. Then
any nowhere vanishing holomorphic section s of ν∗B can be seen as a self-dual
biextension variation of mixed Hodge structure of V on U. We call
h=−logksk
POSITIVITY OF THE HEIGHT JUMP DIVISOR 3
the height of (the biextension variation) s. Its Levi form
i
∂∂¯h
π
coincides with the first Chern form c (ν∗B) of the smooth hermitian line bundle
1
ν∗B on U.
We have the following two important results.
Theorem 1.2. (Hain [12,Theorem13.1],Pearlstein and Peters [21,Theorem8.2],
C. Schnell [24, Lemma 2.2]) The first Chern form c (ν∗B) is semipositive on U.
1
Theorem 1.3. (Lear [18, Theorem 6.6], Hain [12, Theorem 6.1], Pearlstein [22,
Theorem5.19])AssumethatE isareducednormalcrossingsdivisor, thatthevaria-
tionofHodgestructureVisadmissible, andthatν isanadmissiblenormalfunction.
Then the smooth hermitian line bundle ν∗B is Lear-extendable over X. In fact, a
positive tensor power of ν∗B has a continuous extension over X \|E|sing.
We refer to [22] for a discussion of admissibility of variations of mixed Hodge
structures on U, and of normal functions. The admissibility condition on V im-
plies that the monodromy operators of V about the irreducible components of
E are unipotent. A polarized variation of Hodge structures V having unipotent
monodromy and induced by the cohomology of a projective smooth morphism of
algebraicvarieties T →U, and with X ⊃U a smooth algebraicvariety with X\U
a normal crossings divisor, is admissible. In the latter situation, any section ν of
J(V)→U determinedbyafamily ofcyclesonT overU viaGriffiths’sAbel-Jacobi
map is an admissible normal function.
Nowasaboveletϕ: C →X withC aconnectedRiemannsurfacebe aholomor-
phicmapsuchthatthegenericpointofC hasimageinU,andwriteV =ϕ−1U. By
combining Theorems 1.1, 1.2 and 1.3 we immediately obtain the following result.
Theorem 1.4. Assume that E is a reduced normal crossings divisor, that the
variation V on U is admissible, and that the section ν is an admissible normal
function. Then the height jump divisor J on C determined by the smooth
ϕ,ν∗B
hermitian line bundle ν∗B on U and the map ϕ is effective.
A special case of this result related to the so-calledCeresa cycle in the jacobian
of a pointed compact connected Riemann surface of genus g ≥ 2 was conjectured
by Hain [12, Conjecture 14.5]. We will elaborate on this special case in Section 4
below.
Forthespecialcaseofsectionsoffamiliesofprincipallypolarizedabelianvarieties
Theorem 1.4 was shown, among other things, in our previous paper [7]. In this
setting B is the classical Poincar´e bundle. The method of proof in [7] is based
on an explicit formula for the metric on B together with a precise analysis of its
asymptotics. In the even more special case of sections of jacobians, the positivity
of the height jump divisor was obtained in [3], [14]. In these two references, the
positivitywasobtainedbyestablishingtheconcavity,inasuitablesense,ofGreen’s
functionsofelectriccircuits. Notethatnoneofthesepreviousresultsweresufficient
to treat the Ceresa cycle.
A general interpretation of the height jump divisor for admissible biextension
variations of mixed Hodge structures has been given by Brosnan and Pearlstein in
termsoftheirnotionofamonodromybiextensionintheunpublishedmanuscript[4].
4 JOSE´ IGNACIOBURGOSGIL,DAVIDHOLMES,ANDROBINDEJONG
We will actually prove a slight extension of Theorem 1.1 to the case of so-called
subpolynomialLearextensions,cf. Theorem3.2below. Moreoverwewillworkwith
theslightlymoreflexiblenotionofmetrizedR-divisors. Forexample,fromasmooth
hermitian holomorphic line bundle L we obtain a metrized R-divisor by taking the
divisorandheightofanon-zerorationalsection. WedefinethesubpolynomialLear
extension of a metrized R-divisor in Section 2.1.
The structure of this note is as follows. In Section 2 we introduce terminology
and state some preliminary results, including the key Lemma 2.11 which is merely
a local version of a lemma from [2, Chapter XI]. In Section 3 we state and prove
ourmainresult,fromwhichTheorem1.1immediately follows. Finally inSection4
we discuss applications of our main result. We discuss the special case related
to the Ceresa cycle, as well as its ramifications for refined slope inequalities on
M , following [12, Section 14]. Also we discuss the concavity of the local height
g
jump multiplicities and an example related to Arakelov line bundles on families of
compact and connected Riemann surfaces.
2. Preliminary results
2.1. Metrized R-divisors. Ourpurpose isto study extensionsofline bundles (or
Cartier divisors) determined by a smooth hermitian metric. Since a metric is an
analytic object it seems better to allow real coefficients for such extensions and
not to be confined to rationalcoefficients. Now for realcoefficients the language of
Cartier divisors and Green functions is more natural than that of line bundles and
metrics, and thus we adapt this language from now on. For the theory of divisors
on complex manifolds the reader is referred to [15]. In this section, we recall from
[6] the basic definitions concerning metrized R-divisors. We start by recalling the
geometric theory of R-divisors, details of which can be found in [17].
Let X be a complex manifold. A Weil divisor on X is a locally finite Z-linear
combinationof irreducible analytic hypersurfaces of X. An analytic subset of X is
called irreducible if its set of non-singular points is connected. Since X is smooth,
the concepts of Cartier andWeil divisorcoincide. We denote by Car(X)the group
of Cartier divisors of X. The vector space of R-Cartier divisors of X is defined as
Car(X)R =Car(X)⊗ZR.
Thus,anR-CartierdivisoronX isaformallinearcombination α D withα ∈R
i i i i
and Di ∈Car(X). From now on R-Cartier divisors will be calPled R-divisors, while
Cartier divisors will be called divisors. The support of a divisor D is denoted by
|D|. An R-divisor D is said to be ample (respectively effective, nef) if D is an
R-linear combination of ample (respectively effective, nef) divisors with positive
coefficients.
We now introduce metrized divisors and metrized R-divisors.
Definition 2.1. Let D be a Cartier divisor on X. A Green function for D is
a smooth function g: X \|D| → R such that, for each point p ∈ X, there is a
neigbourhoodV of p anda localequationf ofD| suchthat g+log|f| extends to
V
a smooth function on V. A metrized divisor on X is a pair D =(D,g ) where D
D
is adivisoronX andg is aGreenfunction forD. The groupofmetrizeddivisors
D
on X is denoted by Car(X).
d
POSITIVITY OF THE HEIGHT JUMP DIVISOR 5
Remark 2.2. Given a holomorphic line bundle L on X provided with a smooth
hermitian metric k·k and a non-zero meromorphic section s, we obtain a metrized
divisor
div(s)=(div(s),−logksk)
onX. Howeverwe note that the notions of hermitian line bundle with section and
c
ofmetrizeddivisorarenotexactlythesame. Forexample,ifwemultiplythesection
sbyanon-zerocomplexnumberζ andthemetricbythepositiverealnumber|ζ|−1,
the associated metrized divisor does not change.
Definition 2.3. Let X be a complex manifold. The group of metrized R-divisors
on X is the quotient
Car(X)R = Car(X)⊗ZR ∼
.
where ∼ is the equivalencedrelation gidven by α D ∼ β E if and only if
i i i j j j
iαiDi = jβjEj and there is a dense openPsubset U ofPX such that
P P
α g (p)= β g (p) for allp∈U.
i Di j Ej
Xi Xj
Let D = α D . Then the function g = α g : U → R is called the Green
i i i D i i Di
function oPf D. P
Definition 2.4. The first Chern form of a metrized R-divisor D =(D,g ) is
D
i
c (D)= ∂∂¯g .
1 D
π
This is a smooth (1,1)-form on X.
Remark 2.5. For a holomorphic line bundle L on X with smooth hermitian metric
k·k and non-zero meromorphic section s we have c (div(s))=c (L,k·k).
1 1
2.2. Subpolynomial Lear extensions. In this sectcion we introduce subpolyno-
mial Lear extensions of metrized R-divisors. This concept is a mild generalization
ofthe Mumford-Learextensionsintroducedin the contextofmetrized line bundles
in [5] and is suggested by [2, Lemma XI.9.17] where it is proved that, for compact
Riemann surfaces, semipositivity and subpolynomial growth together are enough
tocompute the degreeofa hermitianline bundle usingChern-Weiltheory. We will
crucially use a local version of [2, Lemma XI.9.17] in the proof of our main result.
Definition 2.6. A function µ: R → R is called subpolynomial if, for all n ∈
>0 >0
R ,
>0
µ(x)
lim =0.
x→∞ xn
We will use subpolynomial functions to control the growth of our metrics along
the boundary. A typical example of a subpolynomial function is the logarithm
µ(x)=log(1+x).
Let X be a complex manifold of pure dimension d, and E a reduced divisor on
X. Write U =X \|E|.
Definition 2.7. Let D = (D,g ) be a metrized R-divisor on U. We say that
D
a subpolynomial (sp-) Lear extension of D over X exists if there is an auxiliary
metrized R-divisorD =(D ,g ) on X with D | =D and an analytic subset
X X DX X U
S ⊂ |E| of codimension at least 2 in X, such that for all p ∈ |E|\S there is an
6 JOSE´ IGNACIOBURGOSGIL,DAVIDHOLMES,ANDROBINDEJONG
open neighbourhoodV of p in X\S, a local equation f of E, a realnumber a and
a subpolynomial function µ such that the inequalities
(2.1) −log(µ(1/|f|))≤g −g +alog|f|≤log(µ(1/|f|))
D DX
are satisfied on V ∩U.
Remark 2.8. The realnumbera inthe previousdefinitiondepends onthe choiceof
the auxiliary metrized divisor D and on the point p. However by definition the
X
realnumberaislocallyconstanton|E|\S. Inparticular,ifp,q belongtothesame
irreducible component of E, then the real numbers associated to p,q agree.
Definition 2.9. Let D = (D,g ) be a metrized R-divisor on U. Let {E } be
D i i∈I
the irreducible components of E. Assume that a sp-Lear extension of D over X
exists, with auxiliary metrized divisor D . By Remark 2.8 one has well defined
X
real numbers a , i∈I, one for each irreducible component E of E. We define the
i i
sp-Lear extension of D over X as the R-divisor
(2.2) [D,X]=D + a E .
X i i
Xi∈I
Note that the sum in (2.2) is locally finite, as E is a divisor.
Proposition 2.10. If D has a sp-Lear extension over X, it is unique.
Proof. Let D′X = (DX′ ,gDX′ ) and S′ be a different choice of auxiliary metrized R-
divisor and analytic subset in Definition 2.7. Then there are real numbers {b }
i i∈I
such that
D′ =D + b E .
X X i i
Xi∈I
Without loss of generality we may assume that S = S′ and that |E|sing ⊂ S. Let
p ∈ |E|\S and V a small enough neighborhood of p that only intersects a single
component E of E. Let f′ be another choice of local equation for E on V. Then
i
f′ =uf foranon-vanishingholomorphicfunctionuonV. Finallyletµ′ beanother
choice of subpolynomial function such that
(2.3) −log(µ′(1/|f′|))≤gD−gDX′ +a′ilog|f′|≤log(µ′(1/|f′|)).
After shrinking V if necessary,we can find a third subpolynomialfunction µ′′ such
that
0<log(µ′(1/|f′|))≤log(µ′′(1/|f|)) and 0<log(µ(1/|f|))≤log(µ′′(1/|f|)).
By definition of Green function, there is a smooth function ϕ on V such that
gDX′ =gDX −bilog|f|+ϕ.
Then, subtracting equation (2.1) from equation (2.3) we deduce
−2log(µ′′(1/|f|))≤(b +a′ −a )log|f|+log|u|−ϕ≤2log(µ′′(1/|f|))
i i i
which implies that a =a′ +b . We conclude
i i i
D′ + a′E =D + (b +a′)E =D + a E ,
X i i X i i i X i i
Xi∈I Xi∈I Xi∈I
and this proves the result. (cid:3)
POSITIVITY OF THE HEIGHT JUMP DIVISOR 7
2.3. Residues of semipositive forms. We will write ∆ ⊂ C for the closed unit
disk, and ∆∗ =∆\{0} for the punctured closed disk. For 0<t≤1, we denote by
∆ the closed disk of radius t.
t
Let U be a complex manifold and ω a (1,1)-form on U. Recall that ω is called
semipositive if for every continuous map ϕ: ∆ → U which is holomorphic on the
t
interior of ∆ , the positivity condition
t
ϕ∗ω ≥0
Z
∆t
holds. Fix 0<s≤1. Letg be a smoothfunction onanopenneighbourhoodof∆∗
s
taking values in R. We write
i i
ω := ∂∂¯g, η := ∂¯g.
π π
The next lemma is a localvarianton [2, Lemma XI.9.17], where a globalversionis
given. We give the direct proof of the local version here, following the arguments
in [2], since it only takes little longer than deducing the local case from the global
case. Moreoverthe result is central in our arguments.
Lemma 2.11. Suppose that ω is a semipositive (1,1)-form on an open neighbour-
hood of ∆∗, and that there exist a∈R and a subpolynomial function µ such that
s
(2.4) −log(µ(1/|z|))≤g(z)+alog|z|≤log(µ(1/|z|)).
Then ω :=lim ω exists in R, and we have
∆s ε→0 ∆s\∆ε
R R
0≤ ω = η+a.
Z Z
∆s ∂∆s
Proof. For ease of notationwe will treat only the case s=1. For each0<t≤1, a
computation in polar coordinates yields
i ∂g 1∂g
∂¯g| = − r − dθ.
∂∆t (cid:18) 2 ∂r 2∂θ(cid:19)
We define
−1 ∂g
F(t)=− η = r dθ.
Z 2π Z ∂r
∂∆t ∂∆t
For t <t we have using Stokes’s theorem
1 2
F(t )−F(t )=− η+ η =− ω ≤0.
2 1
Z Z Z
∂∆t2 ∂∆t1 ∆t2\∆t1
Thus, the function F is non-increasing.
Now for any 0<ε≤t≤1 we compute
t dr
F(t)(logt−logε)= F(t)
Z r
ε
t dr
≤ F(r)
Z r
ε
−1 t 2π ∂g
= drdθ
2π Z Z ∂r
ε 0
−1 2π
= g(teiθ)−g(εeiθ)dθ.
2π Z
0
8 JOSE´ IGNACIOBURGOSGIL,DAVIDHOLMES,ANDROBINDEJONG
Fixing t and letting ε vary we find
1 2π
−F(t)logε≤constant+ g(εeiθ)dθ.
2π Z
0
Applying the rightmost bound from formula (2.4) we see that
−F(t)logε≤constant−alog(ε)+log(µ(1/ε)).
The fact that µ is subpolynomial and that, for ε < 1, −logε > 0, implies that
F(t)≤a. In particular we find that the limit
K := lim− η
t→0 Z∂∆t
exists and that K ≤a.
Using again that F is non-increasing, we deduce that −K ≤ −F(t) for all t.
Thus, for 0<ε<t≤1,
t dr 1 2π
−K(logt−logε)≤ −F(r) = g(teiθ)−g(εeiθ)dθ.
Z r 2π Z
ε 0
Fixing t and applying the leftmost bound from formula (2.4) we obtain
Klogε≤constant+alog(ε)+log(µ(1/ε)).
From this we deduce that K ≥a, so that in fact K =a, and proving in particular
that
(2.5) lim η+a=0.
t→0Z∂∆t
Finally we apply Stokes’s Theorem to deduce
0≤ ω = lim ω = η−lim η = η+a.
Z∆ t→0Z∆\∆t Z∂∆ t→0Z∂∆t Z∂∆
This proves the lemma. (cid:3)
Corollary 2.12. Let R={z ,...,z }⊂∆ be a finite set of points with |z |<s,
1 k s i
and let g be a smooth function on an open neighborhood of ∆ \R such that ω =
s
i∂∂¯g is semipositive, and such that there is a subpolynomial function µ and real
π
numbers a such that
i
−log(µ(1/|z−z |))≤g(z)+a log|z−z |≤log(µ(1/|z−z |))
i i i i
when z is close to z . Then the (in)equalities
i
k
0≤ ω = η+ a
i
Z Z
∆s ∂∆s Xi=1
hold, where η = i∂¯g.
π
Proof. It is enough to cut out a small disc around each point z , contained in the
i
interior of ∆ , to apply Lemma 2.11 to each of these discs, and to apply Stokes’s
s
theorem to the complement of the discs. (cid:3)
A metrized R-divisor D = (D,g ) on U is called semipositive if c (D) is a
D 1
semipositive (1,1)-form on U.
POSITIVITY OF THE HEIGHT JUMP DIVISOR 9
Corollary 2.13. Assume that X is a Riemann surface. Let D be a metrized R-
divisor on U = X \|E|. Assume that D is semipositive and that D has a sp-Lear
extension over X. Then c (D) is locally integrable on X. If X is compact, then
1
the (in)equalities
0≤ c (D)=deg[D,X]
1
Z
U
hold.
Proof. We write
i i
ω = ∂∂¯g , η = ∂¯g
D D
π π
on U as before. The local integrability of c (D)= ω follows directly from Lemma
1
2.11. Assume that X is compact. The bound c (D) ≥ 0 is then clear. Let
U 1
p∈|E|. Applying the resultfound in equation(2.R5)to smalldisks ∆ε aroundp we
find that
−lim η =a,
ε→0Z∂∆ε
where a is the local contribution at p to deg[D,X]. The usual argument using
Stokes’s theorem then gives the stated equality. (cid:3)
Remark 2.14. LetU beaconnectedRiemannsurfacewithsmoothcompactification
X, and let V be an admissible polarized variation of Hodge structures of weight
−1 as in the Introduction. Let ν: U → J(V) be an admissible normal function
and denote by B the canonically metrized biextension line bundle on J(V). Us-
ing Theorems 1.2 and 1.3, as a special case of Corollary 2.13 we obtain the local
integrability of the (1,1)-form c (ν∗B), together with the equality
1
c (ν∗B)=deg[ν∗B,X].
1
Z
U
This result proves [12, Conjecture 6.6]. That [12, Conjecture 6.6] holds has previ-
ously been noted by Pearlstein in [20, Corollary 3.2] and by Pearlstein and Peters
in [21, Section 1.5.6].
3. Positivity of the height jump divisor
Let X be a complex manifold, E a reduced divisor on X and U = X \|E|.
Let D = (D,g ) be a metrized R-divisor on U. Let C be a connected Riemann
D
surface and ϕ: C → X a holomorphic map such that the image of the generic
point of C lies in U \|E|. Write V = ϕ−1(U). We have a well defined R-divisor
ϕ|∗D on V and a Green function ϕ|∗g = g ◦ϕ| giving a metrized R-divisor
V V D D V
ϕ|∗D =(ϕ|∗ D,ϕ|∗g ) on V.
V V V D
Definition 3.1. Let X, U, D and ϕ: C →X be as above. Assume that D admits
a sp-Lear extension [D,X] to X. Assume furthermore that the metrized divisor
ϕ|∗D on V admits a sp-Lear extension [ϕ|∗D,C] to C. Then the height jump
V V
divisor is defined as the difference
J =ϕ∗[D,X]−[ϕ|∗D,C],
ϕ,D V
which is an R-divisor on C.
10 JOSE´ IGNACIOBURGOSGIL,DAVIDHOLMES,ANDROBINDEJONG
In particular, the height jump divisor measures the lack of functoriality of a sp-
Learextension. Themainresultofthepresentnoteisthefollowingresult,ofwhich
Theorem 1.1 is an immediate consequence. Note that we are neither assuming X
nor C to be compact, at this point.
Theorem 3.2. Assumethat the hypotheses of Definition 3.1 are satisfied, and that
the metrized divisor D = (D,g ) on U is semipositive, that is, the smooth (1,1)-
D
form c (D) on U is semipositive. Then the height jump divisor J =ϕ∗[D,X]−
1 ϕ,D
[ϕ|∗D,C] is an effective R-divisor on C.
V
Proof. Since we are assuming that D has a sp-Lear extension over X, we have
in particular an auxiliary metrized R-divisor D = (D ,g ) on X such that
X X DX
D | = D. We may assume that D has no common components with E. Next,
X U X
observe that the statement is local on C, and that the support of the height jump
divisor is contained in C \V. Let p ∈ C such that ϕ(p) ∈ |E|. Choose a local
coordinate z in a neighborhood of p such that p is given by z =0, write
B (p)={q ∈C ||z(q)|≤ε}
ε
and choose an ε > 0 small enough such that B (p)∩ϕ−1(|E|∪|D |) = {p} and
ε X
that B (p) is homeomorphic to a closed disk. For δ ≥ 0 and small enough, let
ε
ϕ : B (p)→X be agenericdeformationofϕvaryingcontinuouslywithδ. Bythis
δ ε
we mean that
(1) forallδthemapϕ iscontinuous,andholomorphicontheinteriorofB (p);
δ ε
(2) ϕ =ϕ| ;
0 Bε(p)
(3) for δ > 0, ϕ (B (p)) meets E ∪D properly and transversely and avoids
δ ε X
sing
(|E|∪|D |) ;
X
(4) the number of intersection points of ϕ (B (p)) with |E| (respectively with
δ ε
|D |)isconstantandagreeswiththemultiplicityatpofϕ∗E (respectively
X
of ϕ∗D ).
X,red
LetE ,...,E betheirreduciblecomponentsofE thatmeetasmallneighborhood
1 k
of ϕ(p). We denote by m the multiplicity of ϕ∗D at the point p and by m
X,red i
the multiplicity at p of ϕ∗E , i = 1,...,k. Let a ,...,a be the real numbers
i 1 k
appearing in Remark 2.8 for the sp-Lear extension of D to X and the auxiliary
metrizedR-divisor D and a the one for the sp-Learextension ofϕ∗| D to C at
X C V
p with auxiliary divisor ϕ∗D . Then the multiplicity of the height jump divisor
X
J at p is given by
ϕ,D
k k
m a +m−a −m= m a −a .
i i C i i C
Xi=1 Xi=1
Thus we are reduced to proving that a ≤ m a . We write
C i i
P
i i
ω = ∂∂¯g , η = ∂¯g
D D
π π
on U as usual. From Corollary 2.12 we deduce that for all small δ >0 we have
k
0≤ ϕ∗ω = ϕ∗η+ m a +m.
Z δ Z δ i i
Bε(p) ∂Bε(p) Xi=1